Question

Find the following cardinalities. $$ |\\{\\{1\\},\\{2,\\{3,4\\}\\}, \varnothing\\}| $$

Short answer

The cardinality of the set \(\{\{1\},\{2,\{3,4\}\}, \varnothing\}\) is 3

Step-by-step solution

Identify Elements in the Set

The given set is \(\{\{1\},\{2,\{3,4\}\}, \varnothing\}\). The set contains three elements which are another sets themselves: \(\{1\}\), \(\{2,\{3,4\}\}\) and \(\varnothing\) (the empty set)

Count the Elements

The cardinality of a set is just the number of elements in the set. In this case, there are three elements, hence the cardinality is 3.

Key concepts

Basic Set Theory

Set theory is a foundational system in mathematics that deals with the collection of objects, known as 'elements' or 'members'. The beauty of set theory lies in its versatility, as it can be used to define nearly all mathematical objects.

At its core, a set is defined simply by its elements. These elements could be numbers, characters, or even other sets! The rule is straightforward: if something is considered an element of a set, it is 'in' the set; if not, it is 'out'.

Mathematicians often use curly braces \( \{ \} \) to denote a set and list within them the elements separated by commas. For example, \( \{2, 4, 6 \} \) is a set containing the even numbers 2, 4, and 6.

Some fundamental operations related to sets include union (combining sets), intersection (elements common to all sets), and complement (elements not in the set). However, understanding the basic concept of a set as a collection is the first step to grasping more complex set theory operations.

Cardinality of a Set

The cardinality of a set is a measure of the 'number of elements' in the set. It represents the size or count of the membership of the set. For finite sets, it's quite easy to determine: simply count the number of distinct elements.

For instance, if we have a set \( A = \{ 5, 10, 15 \} \) then the cardinality, denoted \( |A| \) is 3, because there are three elements in the set A.

However, cardinality can become more complex when dealing with infinite sets, like the set of all natural numbers. An infinite set can have a cardinality that signifies the type of infinity it represents, such as countable or uncountable infinity. This extension of the concept of cardinality shows the depth and complexity possible within basic set theory.

Empty Set

The empty set, denoted \( \varnothing \) or \( \{\} \) is a fundamental concept in set theory. It is the unique set that contains no elements. Despite seeming simple, the empty set is quite profound!

It's the foundation for building numerical systems, much like the number zero. The cardinality of the empty set is 0 because there are no elements to count.

Understanding the empty set helps in grasping operations such as the intersection of sets, where the result might be no common elements, or in defining a subset. Any set will always have at least one subset – the empty set.